Working Draft The Ontological Argument Revisited George Cronk Introduction TBA a brief review of the history and the literature and a statement of where this paper fits Conceptual framework Why is the ontological argument called ontological? The expression "ontological argument" was coined by Kant in his Critique of Pure Reason (1781, 1787), but the type of theistic argument in question originated with Anselm of Canturbury in the 11th century (in his Proslogion). Kant called this type of argument "ontological" because it is based on a query as to what kind of being (Greek, ὤν, on; ὄντος, ontos) is referenced by the word "God" (507-8). What kind of a being would God be if God was? According to Anselm, the word God refers to that than which nothing greater can be conceived (Basic Writings 53). The concepts of necessity and contingency Linguistic-logical application: as to statements: Necessarily true statements (tautologies) cannot be false, and necessarily false statements (contradictions) cannot be true. Contingent statements can be either true or false, depending on facts, evidence, & circumstances. 1
Metaphysical (ontological) application: as to beings: Necessary beings (i.e., things with necessary existence) cannot not-exist. Impossible beings (i.e., things whose existence is contradictory, e.g., round squares) cannot exist. Contingent beings are beings whose existence and nonexistence are neither necessary nor impossible). They may exist (rocks) or not (unicorns). Their existence is (logically) possible, and their nonexistence is also (logically) possible. The question that gives rise to the ontological argument for the existence of god: Is it logically possible for the existence of that than which nothing greater can be conceived to be contingent? More precisely, is the nonexistence of that than which nothing greater can be conceived logically possible? Anselm's ontological arguments Anselm s 1st ontological argument: that God exists. The text (Proslogion 263-4): And so, Lord, you who can add understanding to faith, allow me (to the extent that it is good for me) to understand that you exist as I believe you to exist and that you are what I believe you are. I believe you are that than which nothing greater can be conceived [aliquid quo nihil maius cogitari possit]. Is it possible that nothing like that exists? After all, the fool has said in his heart there is no God (Psalms 14:1 and 53:1). But when this fool hears the words that than which nothing greater can be conceived, he must understand what he hears; and what he understands then exists in his mind [in intellectu eius est], even if he doesn t think that such a being exists in fact. For there is a big difference between something existing [as an idea] in someone s mind and... that thing s existing in reality. When a painter first imagines what he is going to paint, he has it in his mind; but, since he has not yet made the painting itself, he doesn t think that it exists yet. Once he has made the painting, however, he not only has the idea of it in his mind, but he also knows that the painting itself exists in fact.... So even a fool would have to admit that that than which nothing greater can be conceived exists [as an idea] in his mind since he understands this phrase when he hears it, and whatever is understood exists at least in the understanding (or mind). But here s my main point: That than which nothing greater can be conceived cannot exist only [as an 2
idea] in the mind because, in addition to existing [as an idea] in the mind, it can also be thought of as existing in reality [that is, objectively], which is greater [than existing only as an idea in the mind]. If that than which nothing greater can be conceived exists only as an idea in the mind, then that than which nothing greater can be conceived is that than which something greater can be conceived, which is impossible [because it is self-contradictory]. Therefore, it necessarily follows that that than which nothing greater can be conceived must exist, not only as an idea in the mind, but in reality. Presentation in (more or less standard) argument form: 1. That than which nothing greater can be conceived cannot exist only as an idea in the mind because, in addition to existing as an idea in the mind, it can also be thought of as existing in reality, that is, objectively, which is greater than existing only as an idea in the mind. 2. If that than which nothing greater can be conceived exists only as an idea in the mind, then that than which nothing greater can be conceived is that than which something greater can be conceived, which is impossible because it is self-contradictory. -------------------------------------------------------------------------------------------------------------- 3. That than which nothing greater can be conceived must exist, not only as an idea in the mind, but in reality. In other words, a God that actually exists is greater than a God that exists only as an idea in the mind. If God exists only as an idea in the mind, then God is not-god (because something that exists only as an idea in the mind is not that than which nothing greater can be conceived ). Thus, the claim that God does not actually exist implies a contradiction and is therefore necessarily false. If the claim that God does not actually exist is necessarily false, then the claim that God actually exists is necessarily true (because the negation of a contradiction is a tautology). So God must exist, and atheism must be false, right? Anselm s 2d ontological argument: that the nonexistence of God is impossible. The text (Proslogion 264): Furthermore, that than which nothing greater can be conceived so certainly exists that it is impossible to think that it doesn t exist. It is possible to think of something that cannot be thought not to exist [that is, a necessary being], and such a being would be greater than something that can be thought not to exist [that is, a contingent being]. If that than which 3
nothing greater can be conceived could be thought of as not existing, then that than which nothing greater can be conceived would not be that than which nothing greater can be conceived, which is an outright contradiction and thus absurd. Therefore, that than which nothing greater can be conceived has such a high degree of existence [that is, necessary existence] that it cannot be thought of as not existing [that is, its nonexistence is impossible]. And you are this being, Oh Lord our God. You exist so truly... that you cannot be even thought of as not existing. And this is appropriate, for if the human mind could conceive of something greater than you, then a creature would rise above its creator and pass judgment on him, which is utterly absurd. Now, everything that exists, other than you alone, can be thought of as not existing. [That is, nothing else than God can be thought of as that than which nothing greater can be conceived.] You alone among all things have the truest and greatest degree of existence [necessary existence]; nothing else has that kind of existence [that is, everything else that exists has only contingent existence]. Presentation in (more or less standard) argument form: 1. It is possible to think of something that cannot be thought not to exist [that is, a necessary being]. 2. A necessary being [something that cannot be thought not to exist] would be greater than something that can be thought not to exist [that is, a contingent being]. 3. If that than which nothing greater can be conceived could be thought of as not existing, then that than which nothing greater can be conceived would not be that than which nothing greater can be conceived, which is an outright contradiction and thus absurd. ------------------------------------------------------------------------------------------------------------- 4. That than which nothing greater can be conceived has such a high degree of existence, that is, necessary existence, that it cannot be thought of as not existing, that is, its nonexistence is impossible. In other words, It is possible to think of a necessary being, i.e., a being whose nonexistence is impossible. Necessary existence is greater than contingent existence, and a necessary being is greater than a contingent being. If the nonexistence of God is possible, then God must be a contingent being. But then God would be not-god because a contingent being cannot be that than which nothing greater can be conceived. Thus, the claim that God s nonexistence is possible implies a contradiction and is therefore necessarily false. 4
If the claim that God s nonexistence is possible is necessarily false, then the claim that God s nonexistence is impossible is necessarily true (because the negation of a contradiction is a tautology). So God s nonexistence is impossible, and therefore God must exist. Thus, agnosticism must be false too, right? Furthermore, God is the only being whose nonexistence is logically impossible. That is, no other being deserves the title of that than which nothing greater can be conceived. The existences of all other beings (actual or conceivable) are either contingent or impossible (Basic Writings 319). So how can the fool doubt or deny the existence of God? Anselm s answer: The fool (i.e., the atheist or agnostic) does not understand the true meaning of that than which nothing greater can be conceived (Proslogion 264). What's wrong with the ontological argument? Both Anselm & Descartes assume that the existence of that than which nothing greater can be conceived is logically possible. Anselm s 1st ontological argument (again): 1. That than which nothing greater can be conceived cannot exist only as an idea in the mind because, in addition to existing as an idea in the mind, it can also be thought of as existing in reality, that is, objectively, which is greater than existing only as an idea in the mind. 2. If that than which nothing greater can be conceived exists only as an idea in the mind, then that than which nothing greater can be conceived is that than which something greater can be conceived, which is impossible because it is self-contradictory. -------------------------------------------------------------------------------------------------------------- 3. That than which nothing greater can be conceived must exist, not only as an idea in the mind, but in reality. 5
Criticism of Anselm s 1st argument: In the 1st premise of his 1st argument, Anselm says that that than which nothing greater can be conceived can be thought of as existing in reality. Is that true? What if the existence of that than which nothing greater can be conceived is logically impossible? Anselm s 2d ontological argument (again): 1. It is possible to think of something that cannot be thought not to exist [that is, a necessary being]. 2. A necessary being [something that cannot be thought not to exist] would be greater than something that can be thought not to exist [that is, a contingent being]. 3. If that than which nothing greater can be conceived could be thought of as not existing, then that than which nothing greater can be conceived would not be that than which nothing greater can be conceived, which is an outright contradiction and thus absurd. ------------------------------------------------------------------------------------------------------------- 4. That than which nothing greater can be conceived has such a high degree of existence, that is, necessary existence, that it cannot be thought of as not existing, that is, its nonexistence is impossible. Criticism of Anselm s 2d argument: In the 3d premise of his 2d argument, Anselm says that thinking that that than which nothing greater can be conceived as not existing is an outright contradiction and thus absurd. Is that true? What if the existence of that than which nothing greater can be conceived is logically impossible? In that case, the statement, that than which nothing greater can be conceived exists, would be necessarily false, and its negation ( that than which nothing greater can be conceived does not exist ) would be necessarily true. Let's consider Descartes' version of the argument, which he sets forth in the fifth meditation in his Meditations on First Philosophy (1641) ("Meditations" 298-9): 6
Descartes' version of the ontological argument: 1. If the nonexistence of God (an infinitely perfect being) were possible, then existence would not be part of God s essence (that is, existence would not be a property of the divine nature). 2. If existence were not part of God s essence (that is, a property of the divine nature), then God would be a contingent (rather than necessary) being. 3. The idea of God as a contingent being (that is, the idea of an infinitely perfect being with contingent rather than necessary existence) is selfcontradictory. 4. It is impossible to think of God as not existing. ------------------------------------------------------------------------------------------------------------ 5. The nonexistence of God is impossible. Criticism of Descartes' version: Like Anselm, Descartes (in his 2d and 4th premises assumes that God s existence must be either necessary or contingent. And since it makes no sense to say that God s existence is contingent (3d premise), Descartes (validly) concludes that the existence of God must be necessary (because His nonexistence is impossible). However, again like Anselm, Descartes does not consider the possibility that God s existence might be impossible. If God s existence is impossible, then His existence is neither contingent nor necessary. If God s existence is impossible, then God cannot exist. To develop this point, let s look at another version of the ontological argument. In his book, Philosophy of Religion (48-49), C. Stephen Evans presents the argument as follows: 1. If a Perfect Being exists, then its existence is necessary. 2. If a Perfect Being does not exist, then its existence is impossible. 3. Either a PB exists or it doesn t. 4. Either the existence of a PB is necessary, or it is impossible. 5. The existence of a PB is not impossible. ---------------------------------------------------------------------------------------------- 6. The existence of a PB is necessary. He formalizes the argument in Propositional Logic notation this way: 7
1. P N 2. P I 3. P P 4. N I 5. I ---------------- 6. N The critical question here: Is Premise 5 true? That is, is it possible that the existence of a Perfect Being is impossible? How might it be argued that the existence of a Perfect Being is impossible? Here's an argument against Premise 5 in the foregoing argument (see Hick, Mackie): 1. If there is dysteleological evil (pointless, purposeless, or meaningless pain, suffering, or disorder), then the existence of a Perfect Being is impossible. 2. There is dysteleological evil. -------------------------------------------------------------------------------------------------------- 3. The existence of a Perfect Being is impossible. The first premise here assumes (reasonably) that the existence of dysteleology of any kind is inconsistent with the existence of a Perfect Being because such a Being would have the power and the will to guarantee that everything else that exists, is teleological (having point, purpose, and meaning). The critical question here: Can the second premise in this argument be known to be either true or false? The answer is "no." It does appear that pointless, purposeless, and meaningless pain, suffering, and disorder does exist; however, isn't it at least logically possible that all of the pain, suffering, and disorder in the world is, in fact, teleological, having point, purpose, and meaning? The answer to the latter question seems to be "yes." Conclusion Assuming that the foregoing assessment is correct, what is the outcome of all this? There is a dispute among rational people as to the existence or non-existence of 8
dysteleological evil. That is, it cannot be known whether dysteleological evil exists or not. That means that it might or might not exist. Since dysteleological evil might exist, and since the existence of dysteleological evil would make the existence of a Perfect Being impossible, it follows that it is possible that the existence of a Perfect Being (God) is impossible (and that the non-existence of a Perfect Being is possible). Therefore, the ontological argument does not prove the existence of God. However, since it cannot be known whether dysteleological evil exists or not, and since it is therefore possible that dysteleological evil does not exist, it follows that it is possible that the existence of a Perfect Being (God) is possible; and as argued above, if the existence of a Perfect Being is possible, then its existence is necessary. So since it is possible that dysteleological evil does not exist, the argument for the impossibility of the existence of a Perfect Being on the basis of the alleged existence of dysteleological evil does not prove the nonexistence of God (and thus also does not prove the truth of atheism). The outcome of all this is that both theism and atheism are in doubt; but neither is ruled out (refuted) by either logic or experience. So one must decide where one stands on the God question by either sticking with agnosticism (which is difficult because agnosticism is most often in practice atheism), or by way of a Kierkegaardian leap of faith one way or the other. Logically, agnosticism wins. Existentially, Kierkegaard wins. 9
Addendum Some Modal Formalizations of the Ontological Argument Modal Logic: A mode or modality is a/the way or manner in which something occurs or is experienced, expressed, or done. There are various systems of Modal Logic. The type of ML utilized in this paper classifies propositions (statements) on the basis of whether they affirm or deny the possibility, impossibility, contingency, necessity, or actuality of their content. "Possible worlds" theory and its applications are also important features of ML. Modal Operators: and p = p is necessarily true p = p is possibly truep = p is actually true Christopher Small s Rendition (interpreting Hartshorne) (Small, Web Version, 6-11) 1. If a perfect being [PB] exists, then it 1. P P Axiom 2 is necessary that a PB exists. 2. If it is possible that a PB does not exist, then it is necessary that it is 2. P P Becker s Postulate 2 (Oskar Becker, 1889-1964) possible that a PB does not exist. 3. Either it is necessary that a PB exists or it is possible that a PB does not 3. P P From 1 and 2 via the Law of Excluded Middle exist. 4. Either it is necessary that a PB exists or it is necessary that it is possible 4. P P From 2 and 3 via Substitution that a PB does not exist. 5. If it is possible that a PB does not 5. P P Contrapositive of 1 exist, then a PB does not exist. 6. It is necessary that if it is possible that a PB does not exist then a PB 6. ( P P) From 5 via Necessitation Postulate does not exist. 7. If it is necessary that it is possible 7. P P From 6 via Modus Tollens that a PB does not exist, then it is necessary that a PB does not exist. 8. Either it is necessary that PB exists or it is necessary that a PB does not 8. P P From 4 and 7 via Substitution exist. 9. It is not necessary that a PB does not exist. 9. P Axiom 1 10. It is necessary that a PB exists. 10. P From 8 and 9 via Disjunctive Syllogism 10
Charles Hartshorne's Rendition (Peter Suber s Restatement) 1. If a perfect being [PB] exists, then it is necessary that a perfect being exists. 2. It is not necessary that a PB does not exist. 3. If it is necessary that a PB exists, then a PB exists. 4. Either it is necessary that a PB exists or it is not necessary that a PB exists. 5. If it is not necessary that a PB exists, then it is necessary that it is not necessary that a PB exists. 6. Either it is necessary that a PB exists or it is necessary that it is not necessary that a PB exists. 7. If it is necessary that it is not necessary that a PB exists, then it is necessary that a PB does not exist. 8. Either it is necessary that PB exists or it is necessary that a PB does not exist. 1. P P A PB cannot exist contingently 2. P Or P The existence of a PB is not impossible 3. P P Modal Axiom: Necessity entails actuality 4. P P Law of Excluded Middle 5. P P Becker s Postulate 1 (Oskar Becker, 1889-1964) 6. P P From 4 and 5 via Substitution 7. P P From 1 via Modus Tollens 8. P P From 6 and 7 via Substitution 9. It is necessary that a PB exists. 9. P From 8 and 2 via Disjunctive Syllogism 10. A PB exists. 10. P From 9 and 3 via Modus Ponens C. Stephen Evans's Simplification of the Argument 1. If a perfect being [PB] exists, then it is necessary that a perfect being exists. 2. If a PB does not exist, then it is necessary that it does not exist (i.e., its existence is impossible). 3. Either a PB exists or a PB does not exist. 4. Either it is necessary that a PB exists or it is necessary that a PB does not exist (i.e., the existence of a PB is either necessary of impossible). 5. It is not necessary that a PB does not exist (i.e., its existence is not 1. P P A PB cannot exist contingently. 2. P P A PB cannot come into existence. 3. P P Law of Excluded Middle 4. P P From 1, 2, and 3 5. P Negation of Disjunct in 4 impossible). 6. It is necessary that a PB exists. 6. P From 4 and 5 via Disjunctive Syllogism 11
Kurt Gödel's Proof C.A. Anderson's Version Definition 1: x is God-like if and only if x has every positive property. Definition 2: A property φ is an essence of entity x if and only if x has φ and φ entails every property x has. Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified (i.e., for every φ, if φ is an essence of x, then necessarily there exists a y such that y has φ). Axiom 1: A property is positive if and only if its negation is not positive. Axiom 2: Any property entailed by a positive property is itself positive. Axiom 3: The property of being God-like is positive. Axiom 4: If a property is positive, then it is necessarily positive. Axiom 5: The property of necessarily existing is a positive property. Theorem 1: If a property is positive, then it is consistent (i.e., possibly exemplified). Corollary 1: The property of being God-like is self-consistent, i.e., possible exemplified. Corollary 2: If x is God-like and has a property, then that property is entailed by the property of being God-like. Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing. Theorem 3: Necessarily, the property of being God-like is exemplified. Dana Scott's Version 1. Either a property or its negation is 1. φ[p(~φ) ~P(φ)] Axiom 1 positive, but not both. 2. A property necessarily implied by a 2. φ ψ[(p(φ) x[φ(x) ψ(x)]) P(ψ)] Axiom 2 positive property is positive. 3. Positive properties are possibly 3. φ[p(φ) xφ(x)] Theorem 1 exemplified. 4. A God-like being possesses all 4. G(x) φ[p(φ) φ(x)] Definition 1 positive properties. 5. The property of being God-like is 5. P(G) Axiom 3 positive. 6. Possibly, God exists. 6. xg(x) Corrolary 7. Positive properties are necessarily 7. φ[p(φ) P(φ)] Axiom 4 positive. 8. An essence of an individual is a 8. φ ess x φ(x) ψ(ψ(x) y(φ(y) ψ(y))) Definition 2 property possessed by it and necessarily implying any of its properties. 9. Being God-like is an essence of any 9. x[g(x) G ess x] Theorem 2 God-like being. 10. Necessary existence of an 10. NE(x) φ[φ ess x yφ(y)] Definition 3 individual is the necessary exemplification of all its essences. 11. Necessary existence is a positive 11. P(NE) Axiom 5 property. 12. Necessarily, God exists 12. xg(x) Theorem 3 12
Alvin Plantinga's "Possible Worlds" Rendition (simplified version by K.E. Himma) Plantinga's argument rests on two key concepts (or definitions of two key properties): Maximal Excellence: A being is maximally excellent in a world W if and only if it is omnipotent, omniscient, and morally perfect in W. Maximal Greatness: A being is maximally great in a world W if and only if it is maximally excellent in every possible world. The argument: 1. The concept of a maximally great being is self-consistent. 2. If 1, then there is at least one logically possible world in which a maximally great being exists. 3. Therefore, there is at least one logically possible world in which a maximally great being exists. 4. If a maximally great being exists in one logically possible world, it exists in every logically possible world. -------------------------------------------------------------------------------------------------------------------------------- 5. Therefore, a maximally great being (that is, God) exists in every logically possible world. 13
Sources / References Anderson, C.A., and M. Gettings. "Gödel's Ontological Proof Revisited." Proceedings of Gödel 96 Conference: Logical Foundations of Mathematics, Computer Science, and Physics: Lecture Notes in Logic 6. Ed. Petr Hájek. Brno, Czech Republic: Springer, 1996. 167-172. Anderson, C. Anthony, "Some Emendations of Gödel's Ontological Proof." Faith and Philosophy, 7:3 (July 1990): 291-303. Anselm, St. Anselm s Basic Writings. Trans. S.W. Deane. La Salle, IL: Open Court, 2nd ed., 1962. Anselm of Canterbury. "Proslogion." Trans. and ed. George Cronk. Readings in Philosophy: Eastern and Western Sources. Ed. George Cronk, et al. Plymouth, MI: Hayden-McNeil, 2d ed., 2004. 263-264. Barnes, Jonathan. The Ontological Argument. London: MacMillan, 1972. Becker, Oskar. "Zur Logik der Modalitäten." Jahrbuch für Philosophie und phänomenologische Forschung, Bd. XI (1930): pp. 497-548. Benzmüller, Christoph, and Bruno Woltzenlogel Paleo. "Formalization, Mechanization and Automation of Gödel's Proof of God's Existence." German Research Foundation. 2013. http://www.logic.at/staff/bruno/papers/2013-godproofabstract.pdf/, 23 Feb. 2014. Broad, C.D. Religion, Philosophy and Psychical Research. New York: Routledge & Kegan Paul, 1953. Descartes, René. "Meditations on First Philosophy." Trans. and ed. George Cronk. Readings in Philosophy: Eastern and Western Sources. Ed. George Cronk, et al. Plymouth, MI: Hayden-McNeil, 2d ed., 2004. 283-306. Descartes, René. Meditations on First Philosophy, With Selections from the Objections and Replies. Ed. John Cottingham. Camvbridge: Cambridge Universiyt Press, 1996. Evans, C. Stephen. Philosophy of Religion. Downers Grove, IL: Intervarsity Press, 1985. Findlay, J.N. God s Existence is Necessarily Impossible. New Essays in Philosophical Theology. Ed. Antony Flew and Alasdair MacIntyre. New York: MacMillan, 1955. Gale, Richard. On the Nature and Existence of God. Cambridge: Cambridge University Press, 1991. Gödel, Kurt, "Ontological proof." Kurt Gödel: Collected Works Vol. 3: Unpublished Essays and Letters. Oxford University Press, 1970. 14
Gödel, Kurt. "Notes in Kurt Gödel s Hand." Logic and Theism: Arguments for and Against Beliefs in God. By John Howard Sobel. Cambridge: Cambridge University Press, 2004. Appendix A, 144-145. Hartshorne, Charles. The Logic of Perfection. LaSalle, IL: Open Court, 1962. Hick, John (1966, 1977). Evil and the God of Love. New York: Palgrave Macmillan, 2010. Himma, K.E. "Anselm: Ontological Argument for God's Existence." The Internet Encyclopedia of Philosophy, ISSN 2161-0002, http://www.iep.utm.edu/ont-arg/#h4/, 23 Feb. 2014. Kant, Immanuel (1781, 1787). Critique of Pure Reason. Trans. Norman Kemp Smith. New York: St. Martin's Press, 1929. Leibniz, Gottfried Wilhelm. New Essays Concerning Human Understanding. Trans. A.G. Langley. Chicago, IL: Open Court Publishing, 1896. Mackie, J.L. Evil and Omnipotence. Mind 64(1955): 200-212. Mackie, J.L. The Miracle of Theism: Arguments for and against the Existence of God. Oxford: Oxford University Press, 1982. Malcolm, Norman. Anselm s Ontological Arguments. Philosophical Review, 69:1 (1960), 41-62. Miller, Ed L. God and Reason. Upper Saddle River, NJ: Prentice-Hall, Inc., 2nd Ed., 1995. Oppy, Graham. "Gödelian Ontological Arguments." Analysis 56:4 (October 1996): 226-230. Pike, Nelson. Divine Omniscience and Voluntary Action. Philosophical Review, 74 (1965): 27-46. Plantinga, Alvin. God, Freedom, and Evil. New York: Harper and Row, 1974. Plantinga, Alvin. The Nature of Necessity. Oxford: Clarenden Press, 1974. Plantinga, Alvin, ed. The Ontological Argument from St. Anselm to Contemporary Philosophers. Garden City, NY: Doubleday, 1965. Pojman, Louis. Philosophy of Religion. London: Mayfield Publishing Co., 2001. Rowe, William. Modal Versions of the Ontological Argument. Philosophy of Religion. Ed. Louis Pojman. Belmont, CA: Wadsworth Publishing Co., 3d Ed., 1998. 15
Scott, Dana. "Notes in Dana Scott s Hand." Logic and Theism: Arguments for and Against Beliefs in God. By John Howard Sobel. Cambridge: Cambridge University Press, 2004. Appendix B, 145-146. Sennett, James F. Universe Indexed Properties and the Fate of the Ontological Argument. Religious Studies, 27 (1991): 65-79. Small, C.G. "Reflections on Gödel's Ontological Argument." Klarheit in Religionsdingen: Aktuelle Beitrage zur Religionsphilosophie, Grundlagenprobleme unserer Zeit, Band III. Ed. W. Deppert and M. Rahnfeld. Leipsig: Leipziger Universitatsverlag, 2003. 109-144. Also published at http://sas.uwaterloo.ca/~cgsmall/godel.final.revision.pdf. Sobel, J.H. Logic and Theism: Arguments for and Against Beliefs in God. Cambridge: Cambridge University Press, 2004. Suber, Peter, "The Ontological Argument." Electronic hand-out for Prof. Suber's course on "Rationalism & Empiricism." http://legacy.earlham.edu/~peters/courses/re/onto-arg.htm, 23 Feb 2014. 16